Buying a quota for one 1 unit of catch this season
\(\Pi\): unit profits expected; \(\lambda\) : quota price
\[ \text{Revenue from Buying}= \begin{cases} \Pi - \lambda,& \text{if quota is used} \\ -\lambda, & \text{otherwise} \end{cases} \]
\[ E[\text{Value of Quota}] = 0 \]
\[ \lambda^* = \text{Pr}(\text{Needed})\Pi \]
\[ \begin{aligned} q &= \text{Quota owned} \\ c &= \text{Daily catch} \\ t &= \text{Day of the season} \\ T &= \text{Season length} \end{aligned} \] Then the probability that you will need that unit of quota is just: \[ 1 - \text{Pr}(c \leq \frac{q}{T-t}) \]
For every unit of species 1 I expect to catch \(x_2\) units of species 2
\[ \text{Revenue from Buying}= \begin{cases} \Pi_1 - \lambda_1 + x_2(\Pi_2 - \lambda_2),& \text{needed} \\ -\lambda_1, & \text{otherwise} \end{cases} \]
\[ \lambda_1^* = \text{Pr}(\text{Needed})\left(\Pi_1 + x_2(\Pi_2 - \lambda_2) \right) \]
\[ \text{Score} = \text{Blue Biomass}_{t=20} \]
\[ \text{Score} = \text{Blue Biomass}_{t=20} + \sum_{i=1}^{20} \text{Red Landings}_{t=i}\]
\[ \text{Score} = \text{Blue Biomass}_{t=20} + \sum_{i=1}^{20} \text{Red Landings}_{t=i}\]
In a scenario where fishers are unable to respond to incentives the optimal quotas under TACs and ITQs are exactly the same
A better understanding of some of the trade-offs, particularly that between catch and catch variation, can be achieved by ‘real-time gaming’ of the MSE, which involves the decision-makers managing simulated populations where they are provided with the data which would actually be available on an annual basis
| Method | 20 Years | 80 Years |
|---|---|---|
| Quota - optimized 20 years | 412,056 | 390,581 |
| Biomass controller | 352,566 | 1,058,428 |
| Random controller | 398,069 | 390,678 |
| Anarchy | 230,225 | 202,231 |
| Method | 20 Years | 80 Years |
|---|---|---|
| Quota - optimized 20 years | 412,056 | 390,581 |
| Biomass controller | 352,566 | 1,058,428 |
| Random controller | 398,069 | 390,678 |
| Anarchy | 230,225 | 202,231 |
| Cash-distance controller | 326,116 | 1,001,269 |